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Synthese

, Volume 191, Issue 10, pp 2281–2299 | Cite as

Towards transfinite type theory: rereading Tarski’s Wahrheitsbegriff

  • Iris Loeb
Article

Abstract

In his famous paper Der Wahrheitsbegriff in den formalisierten Sprachen (Polish edition: Nakładem/Prace Towarzystwa Naukowego Warszawskiego, wydzial, III, 1933), Alfred Tarski constructs a materially adequate and formally correct definition of the term “true sentence” for certain kinds of formalised languages. In the case of other formalised languages, he shows that such a construction is impossible but that the term “true sentence” can nevertheless be consistently postulated. In the Postscript that Tarski added to a later version of this paper (Studia Philosophica, 1, 1935), he does not explicitly include limits for the kinds of language for which such a construction is possible. This absence of such limits has been interpreted as an implied claim that such a definition of the term “true sentence” can be constructed for every language. This has far-reaching consequences, not least for the widely held belief that Tarski changed from an universalistic to an anti-universalistic standpoint. We will claim that the consequence of anti-universalism is unwarranted, given that it can be argued that the Postscript is not in conflict with the existence of limits outside of which a definition of “true sentence” cannot be constructed. Moreover, by a discussion of transfinite type theory, we will also be able to accommodate other of the changes made in Tarski’s Postscript within a type-theoretical framework. The awareness of transfinite type theory afforded by this discussion will lead, in turn, to an account of Tarski’s Postscript that shows a gradual change in his logical work, rather than any of the more radical transitions which the Postscript has been claimed to reflect.

Keywords

Tarski Truth Type theory Universalism 

Notes

Acknowledgments

I would like to thank Monika Gruber for discussions on the Wahrheitsbegriff that have been very helpful to me, in particular because in general we seem to disagree. Furthermore I like to thank the participants of the “Tarski Seminar”, which took place in fall term 2011 and made reading Tarski’s paper a very pleasant activity. Arianna Betti, Hein van den Berg, Lieven Decock, Wim de Jong, Jeroen de Ridder, Stefan Roski and Jeroen Smid have read an earlier version of this paper and given me valuable feedback. I thank the anonymous referees for their suggestions. Work on this paper was made possible by ERC Starting Grant TRANH 203194.

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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesVU UniversityAmsterdamThe Netherlands

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